Fact: If a whole number, d, goes into a product, ab, and if
the greatest common factor of a and d is 1, then d goes into b.

Proof: Suppose that d goes into ab. Say ab = md. If the
greatest common factor of a and d is 1, then we can express 1 as a
difference of multiples of a and d. Say that 1 = ra + sd where r and
s are integers. Naturally, one of r and s will be negative and the
other will be positive, but if we assume that they are integers, we
will not have to worry about which one is positive and which one is
negative. Then

b = 1.b

= (ra + sd)b

= rab + sdb

But we just said that ab = md. If we substitute that in we get

b = rmd + sbd

= (rm + sb)d

and we see that b is a multiple of d.

Definition: If the greatest common factors of two numbers
is 1, then we say that the two numbers are relatively prime.

Note that the result need not be true if we drop the condition
that d be relatively prime to one of the factors. For instance 6 goes
into 24 which is 3x8, but 6 does not go into either 3 or 8. The
reason that the last fact does not apply is because 6 is not
relatively prime to either 6 or 8.